Moduli of curves as moduli of A∞-structures

Abstract

We define and study the stack ${\mathcal{U}}_{g,g}^{ns,a}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural ${\mathbb{G}}_m^g$-torsor ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ over ${\mathcal{U}}_{g,g}^{ns,a}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$-invariant of an elliptic curve to the case $g>1$. Our main result is that in characteristics different from $2$ and $3$ the moduli space ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ is isomorphic to the moduli space of minimal $A_{\infty }$-structures on a certain finite-dimensional graded associative algebra $E_g$ (introduced by Fisette and Polishchuk).